$\log_{b^y}{m^x}$ $\,=\,$ $\Big(\dfrac{x}{y}\Big)\log_{b}{m}$

The double power rule of logarithms is a mathematical identity which is used to find the value of logarithm of a quantity by expressing quantity and base quantity of logarithmic term in exponential notation.

$p$ and $q$ are two quantities and assume they are expressed in exponential form as $m^{\displaystyle x}$ and $n^{\displaystyle y}$ respectively.

$p \,=\, m^{\displaystyle x}$ and $q \,=\, n^{\displaystyle y}$

The value of logarithm of $p$ to $q$ is written as $\log_{q}{p}$ in mathematics. Actually, $p \,=\, m^{\displaystyle x}$ and $q \,=\, n^{\displaystyle y}$.

Therefore, $\log_{q}{p}$ $\,=\,$ $\log_{n^y}{m^{\displaystyle x}}$

Take $t \,=\, b^y$ and the logarithmic function can be written as follows.

$\implies \log_{b^y}{m^{\displaystyle x}}$ $\,=\,$ $\log_{t}{m^{\displaystyle x}}$

According to Power law of Logarithms, the log of a quantity in exponential form to a base is equal to the product of exponent and log of the base of exponential term to same base.

$\implies \log_{b^y}{m^{\displaystyle x}}$ $\,=\,$ $x\log_{t}{m}$

Now, replace the actual value of the base $t$.

$\,\,\, \therefore \,\,\,\,\,\, \log_{b^y}{m^{\displaystyle x}}$ $\,=\,$ $x\log_{b^y}{m}$

It is time to find the value of log of $m$ to a base which is expressed in exponential form as $b^{y}$. It can be done by the base power rule of logarithm.

$\,\,\, \therefore \,\,\,\,\,\, \log_{b^y}{m}$ $\,=\,$ $\Big(\dfrac{1}{y}\Big)\log_{b}{m}$

Now, recollect the results of above two steps once.

$(1) \,\,\,\,\,\,$ $\log_{b^y}{m^{\displaystyle x}}$ $\,=\,$ $x\log_{b^y}{m}$

$(2) \,\,\,\,\,\,$ $\log_{b^y}{m}$ $\,=\,$ $\Big(\dfrac{1}{y}\Big)\log_{b}{m}$

Now, combine both results to a log property to find the value of log of a quantity in exponential form to base in exponential form.

$\implies \log_{b^y}{m^x}$ $\,=\,$ $x \times \log_{b^y}{m}$

$\implies \log_{b^y}{m^x}$ $\,=\,$ $x \times \Big(\dfrac{1}{y}\Big)\log_{b}{m}$

$\,\,\, \therefore \,\,\,\,\,\, \log_{b^y}{m^x}$ $\,=\,$ $\Big(\dfrac{x}{y}\Big)\log_{b}{m}$

Thus, the double power rule of logarithms is derived in algebraic form and it can be used as an identity in mathematics.

Latest Math Topics

Jan 06, 2023

Jan 03, 2023

Jan 01, 2023

Dec 26, 2022

Dec 08, 2022

Latest Math Problems

Jan 31, 2023

Nov 25, 2022

Nov 02, 2022

Oct 26, 2022

A best free mathematics education website for students, teachers and researchers.

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Learn how to solve the maths problems in different methods with understandable steps.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved